Adrian Marino

Claudio Collado

Inicialización

Fijamos la semilla para poder reproducir los experimentos. También se fija el numero de CPU’s a utilizar.

set.seed(42)
options(mc.cores = 24)

Librerias

Se importan las librerías a utilizar a lo largo de la notebook:

# install.packages(pacman)
# install.packages("https://cran.r-project.org/src/contrib/rstan_2.21.2.tar.gz",repos = NULL,type="source")
# sudo apt-get install libglpk-dev
library(pacman)
p_load(tidyverse, tidymodels, rsample, rstan, shinystan, rstanarm, devtools)
p_load_gh('adrianmarino/commons')

import('../src/dataset.R')
## [1] "-> '../src/dataset.R' script loadded successfuly!"
import('../src/plot.R')
## [1] "-> '../src/plot.R' script loadded successfuly!"
import('../src/model.R')
## [1] "-> './bayesian_regression_predictor.R' script loadded successfuly!"
## [1] "-> '../src/model.R' script loadded successfuly!"

Dataset y Analisis Exploratorio

Palmer Penguins

Palmer Penguins

1. Lectura del dataset

dataset <- load_dataset() %>% mutate_if(is.character, as.factor)

dataset %>% glimpse()
## Rows: 344
## Columns: 8
## $ species           <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adel…
## $ island            <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgerse…
## $ bill_length_mm    <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, …
## $ bill_depth_mm     <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, …
## $ flipper_length_mm <dbl> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186…
## $ body_mass_g       <dbl> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, …
## $ sex               <fct> male, female, female, NA, female, male, female, male…
## $ year              <dbl> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…

2. Variables

Se enumeran y describen breve-mente cada variable que forma parte del dataset:

Variables Numéricas:

  • bill_length_mm: Longitud del pico del individuo medida en milímetros (también conocida como longitud del culmen).
  • bill_depth_mm: Profundidad del pico del individuo medida en milímetros (también conocida como profundidad del culmen).
  • flipper_length_mm: Longitud de la aleta del individuo medida en milímetros.
  • body_mass_g: Masa corporal del individuo medida en gramos.
  • year: Año en el que se registra el individuo.

Variables Categóricas:

  • species: Especie del individuo (Adelie, Gentoo ;) o Chinstrap).
  • sex: Sexo del individuo.
  • island: Isla donde se encontré el individuo (Biscoe, Dream o Torgersen).

A continuación veamos los posibles valores de las variables categóricas:

show_values(dataset %>% select_if(negate(is.numeric)))
## _____________ 
## species     n 
## ============= 
## Adelie    152 
## Chinstrap  68 
## Gentoo    124 
## ¯¯¯¯¯¯¯¯¯¯¯¯¯ 
## _____________ 
## island      n 
## ============= 
## Biscoe    168 
## Dream     124 
## Torgersen  52 
## ¯¯¯¯¯¯¯¯¯¯¯¯¯ 
## __________ 
## sex      n 
## ========== 
## female 165 
## male   168 
## NA      11 
## ¯¯¯¯¯¯¯¯¯¯

3. Resumen de faltantes

missings_summary(dataset)

4. Varibles numericas

hist_plots(dataset)

Observaciones

  • Se aprecia que cada año se registro prácticamente el mismo numero de individuos.
  • La distribución de la masa corporal de los individuos tiene una asimétrica positiva. Tenemos muchos individuos con valores bajos de masa corporal, con una media de 192 gramos. Luego tenemos menos individuos con valores mas alto.
  • La longitud de la aleta parece ser una distribución bi-modal. Tenemos dos modas una 192 mm y otra en 215 mm.
  • La longitud del pico también parece tener una ligera simetría positiva. Es decir que lo individuos con menor peso tiene pico mas pequeños.
  • Por otro lado la profundidad de pico parece tener una ligera simetría positiva.
box_plots(dataset)

Observaciones

** COMPLETAR

Outliers

No se registran valores mas extremos que el mínimo y máximo valor en cada variables. Es decir que no encontramos outliers.

outliers(dataset, column='bill_length_mm')
## $inf
## [1] 32.1
## 
## $sup
## [1] 59.6
outliers(dataset, column='bill_length_mm')
## $inf
## [1] 32.1
## 
## $sup
## [1] 59.6
outliers(dataset, column='bill_depth_mm')
## $inf
## [1] 13.1
## 
## $sup
## [1] 21.5
outliers(dataset, column='flipper_length_mm')
## $inf
## [1] 172
## 
## $sup
## [1] 231
outliers(dataset, column='body_mass_g')
## $inf
## [1] 2700
## 
## $sup
## [1] 6300
outliers(dataset, column='year')
## $inf
## [1] 2007
## 
## $sup
## [1] 2009
bar_plots(dataset)

Observaciones

  • La variable sexo se encuentra balanceada. Por otro lado, contiene algunos valores faltantes.
  • La variable island esta completamente desbalanceada. Esto seguramente se debe a una diferencia en numero en las poblaciones en cada isla o a un sesgo al momento de registrar los individuos. Es decir que registramos con individuos en una isla que en otra.
  • Lo mismo sucede con las especies de individuos. Vemos un gran desbalance entre la especie Chinstrap vs. otra especies. Por otro aldo Adelie y Gentoo tiene un conteo mas cercano

5. Excluir observaciones con missings

dataset <- dataset %>% drop_na()
missings_summary(dataset)
## Warning: attributes are not identical across measure variables;
## they will be dropped

6. Correlaciones

corr_plot(dataset %>% dplyr::select(-year))

segmented_pairs_plot(dataset, segment_column='species')

Experimentos

Experimento 1

  • Solo variables numéricas
  • Regresión múltiple frecuentista.
  • regresión múltiple bayesiana con priors normales y exponencial.

1. Split train - test

train_test <- train_test_split(dataset, train_size = 0.7, shuffle = TRUE)
## [1] "Train set size: 233"
## [1] "Test set size: 100"
train_set <- train_test[[1]]
test_set  <- train_test[[2]]

2. Modelo lineal

lineal_model_1 <- lm(
  body_mass_g ~ bill_length_mm + bill_depth_mm + flipper_length_mm,
  data = train_set
)

3. Modelo bayesiano

bayesion_model_1 <- stan(
  model_code =  "
    data {
      int<lower=1>               obs_count;
      vector<lower=1>[obs_count] x1;
      vector<lower=1>[obs_count] x2;
      vector<lower=1>[obs_count] x3;
      vector[obs_count]          y;
    }
    parameters {
      real          beta0;
      real          beta1;
      real          beta2;
      real          beta3;
      real<lower=0> sigma;
    }
    model {
      beta0 ~ normal(0, 8000);
      beta1 ~ normal(0, 100);
      beta2 ~ normal(0, 100);
      beta3 ~ normal(0, 100);
      sigma ~ exponential(0.1);
    
      y ~ normal(beta0 + beta1 * x1 + beta2 * x2 + beta3 * x3, sigma);
    }
  ",
  data = list(
      obs_count = nrow(train_set),
      y  = colvalues(train_set, 'body_mass_g'),
      x1 = colvalues(train_set, 'bill_length_mm'),
      x2 = colvalues(train_set, 'bill_depth_mm'),
      x3 = colvalues(train_set, 'flipper_length_mm')
  ),
  chains = 3,
  iter   = 300,
  warmup = 180,
  thin   = 1
)
## Running /usr/local/lib/R/bin/R CMD SHLIB foo.c
## gcc -I"/usr/local/lib/R/include" -DNDEBUG   -I"/usr/local/lib/R/site-library/Rcpp/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/unsupported"  -I"/usr/local/lib/R/site-library/BH/include" -I"/usr/local/lib/R/site-library/StanHeaders/include/src/"  -I"/usr/local/lib/R/site-library/StanHeaders/include/"  -I"/usr/local/lib/R/site-library/RcppParallel/include/"  -I"/usr/local/lib/R/site-library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fpic  -g -O2 -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g  -c foo.c -o foo.o
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:88,
##                  from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
##   628 | namespace Eigen {
##       | ^~~~~~~~~
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
##   628 | namespace Eigen {
##       |                 ^
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
##    96 | #include <complex>
##       |          ^~~~~~~~~
## compilation terminated.
## make: *** [/usr/local/lib/R/etc/Makeconf:168: foo.o] Error 1
params_1 <- c('beta0', 'beta1', 'beta2', 'beta3', 'sigma')
traceplot(bayesion_model_1, pars = params_1, inc_warmup = TRUE)

4. Coeficientes

lm_vs_br_coeficients(lineal_model_1, bayesion_model_1, params_1)

4. Validación

vars_1 <- c('bill_length_mm', 'bill_depth_mm', 'flipper_length_mm') 

models_validation(lineal_model_1, bayesion_model_1, params_1, vars_1, test_set)
bayesion_predictor_1 <- BayesianRegressionPredictor.from(bayesion_model_1, params_1, vars_1)

plot_compare_fit(
  lineal_model_1, 
  bayesion_predictor_1, 
  train_set,
  label_1='Regresion Lineal', 
  label_2='Regresion Bayesiana'
)

Experimento 2

  • Igual al experimento A, incorporar una variable categorica.
  • Regresión multiple frecuentista.
  • Regresion bayesiana con priors normales y exponencial.

1. Modelo lineal

lineal_model_2 <- lm(
    body_mass_g 
      ~ bill_length_mm
      + bill_depth_mm
      + flipper_length_mm
      + sex,

  data = train_set
)

2. Modelo bayesiano

Antes que anda transformamos la columna categórica a un one-hot encoding:

cat_cols      <- c('sex')

train_set_cat <- train_set %>% dummify(cat_cols)
test_set_cat  <- test_set  %>% dummify(cat_cols)

train_set_cat

Construimos una matriz con todas las variables x + intercept.

to_model_input <- function(df) {
  model.matrix(
    body_mass_g 
      ~ bill_length_mm
      + bill_depth_mm
      + flipper_length_mm
      + sex_female
      + sex_male,

    data = df
  )
}

train_X <- train_set_cat %>% to_model_input()
test_X  <- test_set_cat %>% to_model_input()
bayesion_model_2 <- stan(
  model_code = "
    data {
      int<lower=1>                 obs_count;
      int<lower=1>                 coef_count;
      matrix[obs_count,coef_count] X;
      vector[obs_count]            y;
    }
    parameters {
      vector[coef_count]  beta;
      real<lower=0>       sigma;
    }
    model {
      beta[1] ~ normal(0, 2000);
      
      beta[2] ~ normal(0, 30);
      beta[3] ~ normal(0, 100);
      beta[4] ~ normal(0, 100);
  
      beta[5] ~ normal(0, 100);
      beta[6] ~ normal(0, 100);
  
      sigma ~ exponential(0.1);
  
      y ~ normal(X * beta, sigma);
    }
  ",
  data = list(
      obs_count  = dim(train_X)[1],
      coef_count = dim(train_X)[2],
      y          = colvalues(train_set_cat, 'body_mass_g'),
      X          = train_X
  ),
  chains = 3,
  iter   = 300,
  warmup = 180,
  thin   = 1
)
## Running /usr/local/lib/R/bin/R CMD SHLIB foo.c
## gcc -I"/usr/local/lib/R/include" -DNDEBUG   -I"/usr/local/lib/R/site-library/Rcpp/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/unsupported"  -I"/usr/local/lib/R/site-library/BH/include" -I"/usr/local/lib/R/site-library/StanHeaders/include/src/"  -I"/usr/local/lib/R/site-library/StanHeaders/include/"  -I"/usr/local/lib/R/site-library/RcppParallel/include/"  -I"/usr/local/lib/R/site-library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fpic  -g -O2 -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g  -c foo.c -o foo.o
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:88,
##                  from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
##   628 | namespace Eigen {
##       | ^~~~~~~~~
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
##   628 | namespace Eigen {
##       |                 ^
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
##    96 | #include <complex>
##       |          ^~~~~~~~~
## compilation terminated.
## make: *** [/usr/local/lib/R/etc/Makeconf:168: foo.o] Error 1
params_2 <- c('beta[1]', 'beta[2]', 'beta[3]', 'beta[4]', 'beta[5]', 'beta[6]', 'sigma')
traceplot(bayesion_model_2, inc_warmup = TRUE, pars = params_2)

3. Coeficientes

lineal_model_2$coefficients
##       (Intercept)    bill_length_mm     bill_depth_mm flipper_length_mm 
##     -1922.6152066        -0.5330962       -92.9216684        37.2016333 
##           sexmale 
##       534.1583252
br_coeficients(bayesion_model_2, params_2)

4. Validación

vars_2 <- c('bill_length_mm', 'bill_depth_mm', 'flipper_length_mm', 'sex_female','sex_male')

models_validation(
  lineal_model_2, 
  bayesion_model_2, 
  params_2,
  vars_2,
  test_set, 
  test_set_cat
)
bayesion_predictor_2 <- BayesianRegressionPredictor.from(bayesion_model_2, params_2, vars_2)

plot_compare_fit(
  lineal_model_2, 
  bayesion_predictor_2, 
  test_set, 
  test_set_cat,
  label_1='Regresion Lineal', 
  label_2='Regresion Bayesiana'
)

Experimento 3

  • Igual al experimento A incorporando outliers en alguna variable numérica.
  • Regresión múltiple frecuentista.
  • Regresión bayesiana con priors normales y exponencial.

1. Outliers

plot_data(train_set)

train_set_with_outliers <- train_set %>%
  mutate(flipper_length_mm = ifelse(
    body_mass_g > 5600 , 
    flipper_length_mm + (flipper_length_mm * runif(1, 0.2, 0.3)), 
    flipper_length_mm
  ))

plot_data(train_set_with_outliers)

2. Modelo lineal

lineal_model_3 <- lm(
  body_mass_g ~ bill_length_mm + bill_depth_mm + flipper_length_mm,
  data = train_set_with_outliers
)
plot_compare_fit(
  lineal_model_1, 
  lineal_model_3, 
  train_set,
  label_1='Regresión Lineal SIN outliers', 
  label_2='Regresión Lineal CON outliters'
)

lm_vs_lm_coeficients(lineal_model_1, lineal_model_3)

3. Modelo bayesiano

bayesion_model_3 <- stan(
  model_code =  "
    data {
      int<lower=1>               obs_count;
      vector<lower=1>[obs_count] x1;
      vector<lower=1>[obs_count] x2;
      vector<lower=1>[obs_count] x3;
      vector[obs_count]          y;
    }
    parameters {
      real          beta0;
      real          beta1;
      real          beta2;
      real          beta3;
      real<lower=0> sigma;
    }
    model {
      beta0 ~ normal(0, 8000);
      beta1 ~ normal(0, 100);
      beta2 ~ normal(0, 100);
      beta3 ~ normal(0, 100);
      sigma ~ exponential(0.1);
    
      y ~ normal(beta0 + beta1 * x1 + beta2 * x2 + beta3 * x3, sigma);
    }
  ",
  data = list(
      obs_count = nrow(train_set_with_outliers),
      y  = colvalues(train_set_with_outliers, 'body_mass_g'),
      x1 = colvalues(train_set_with_outliers, 'bill_length_mm'),
      x2 = colvalues(train_set_with_outliers, 'bill_depth_mm'),
      x3 = colvalues(train_set_with_outliers, 'flipper_length_mm')
  ),
  chains = 3,
  iter   = 300,
  warmup = 180,
  thin   = 1
)

params_3 <- c('beta0', 'beta1', 'beta2', 'beta3', 'sigma')
traceplot(bayesion_model_3, pars = params_3, inc_warmup = TRUE)

4. Coeficientes

lm_vs_br_coeficients(lineal_model_3, bayesion_model_3, params_3)

5. Validación

vars_3 <- c('bill_length_mm', 'bill_depth_mm', 'flipper_length_mm') 

models_validation(lineal_model_3, bayesion_model_3, params_3, vars_3, test_set)
bayesion_predictor_3 <- BayesianRegressionPredictor.from(bayesion_model_3, params_3, vars_3)

plot_compare_fit(
  lineal_model_1, 
  bayesion_predictor_3, 
  train_set,
  label_1='Regresion Lineal SIN outliers', 
  label_2='Regresion Bayesiana CON outliers'
)

Experimento 4

  • Idem experimento 1 pero reduciendo la cantidad de observaciones a pocos valores (ej:30).
  • Regresion multiple frecuentista.
  • Regresion bayesiana con priors normales y exponencial.

1. Split train - test

En este aso entrenamos solo con el 10% de lo datos.

train_test <- train_test_split(dataset, train_size = 0.05, shuffle = TRUE)
## [1] "Train set size: 16"
## [1] "Test set size: 317"
train_set_4 <- train_test[[1]]
test_set_4  <- train_test[[2]]
plot_data(train_set_4)

2. Modelo lineal

lineal_model_4 <- lm(
  body_mass_g ~ bill_length_mm + bill_depth_mm + flipper_length_mm,
  data = train_set_4
)

3. Modelo bayesiano

bayesion_model_4 <- stan(
  model_code =  "
    data {
      int<lower=1>               obs_count;
      vector<lower=1>[obs_count] x1;
      vector<lower=1>[obs_count] x2;
      vector<lower=1>[obs_count] x3;
      vector[obs_count]          y;
    }
    parameters {
      real          beta0;
      real          beta1;
      real          beta2;
      real          beta3;
      real<lower=0> sigma;
    }
    model {
      beta0 ~ normal(0, 8000);
      beta1 ~ normal(0, 100);
      beta2 ~ normal(0, 100);
      beta3 ~ normal(0, 100);
      sigma ~ exponential(0.1);
    
      y ~ normal(beta0 + beta1 * x1 + beta2 * x2 + beta3 * x3, sigma);
    }
  ",
  data = list(
      obs_count = nrow(train_set_4),
      y  = colvalues(train_set_4, 'body_mass_g'),
      x1 = colvalues(train_set_4, 'bill_length_mm'),
      x2 = colvalues(train_set_4, 'bill_depth_mm'),
      x3 = colvalues(train_set_4, 'flipper_length_mm')
  ),
  chains = 3,
  iter   = 300,
  warmup = 180,
  thin   = 1
)

params_4 <- c('beta0', 'beta1', 'beta2', 'beta3', 'sigma')
traceplot(bayesion_model_4, pars = params_4, inc_warmup = TRUE)

4. Coeficientes

Coeficientes de la regresión múltiple:

lineal_model_4$coefficients
##       (Intercept)    bill_length_mm     bill_depth_mm flipper_length_mm 
##       -8926.26327          24.74177          81.05387          52.87000

Coeficientes descubiertos por la regresión múltiple bayesiana:

for(param in params_4) print(get_posterior_mean(bayesion_model_4, par=param)[4])
## [1] -7895.822
## [1] 26.30672
## [1] 53.77526
## [1] 49.6672
## [1] 236.508

5. Validación

vars_4 <- c('bill_length_mm', 'bill_depth_mm', 'flipper_length_mm') 

models_validation(lineal_model_4, bayesion_model_4, params_4, vars_4, test_set)
bayesion_predictor_4 <- BayesianRegressionPredictor.from(bayesion_model_4, params_4, vars_4)

plot_compare_fit(
  lineal_model_4,
  bayesion_predictor_4, 
  train_set,
  label_1='Regresion Lineal', 
  label_2='Regresion Bayesiana'
)

Experimento 5

  • Igual al experimento 1 pero proponiendo dos nuevas regresiones bayesianas con priors para los parámetros que sean:
    • Una poca informativa (uniforme).
    • Una muy informativa (sesgada o con muy poca varianza).
  • Comparar con resultados de la bayesiana del experimento A

1. Modelo bayesiano con parametro con distribucion poco informativa

1. Modelo

Definimos una distribución uniforme para el beta asociado a la variable flipper_length_mm.

## Running /usr/local/lib/R/bin/R CMD SHLIB foo.c
## gcc -I"/usr/local/lib/R/include" -DNDEBUG   -I"/usr/local/lib/R/site-library/Rcpp/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/"  -I"/usr/local/lib/R/site-library/RcppEigen/include/unsupported"  -I"/usr/local/lib/R/site-library/BH/include" -I"/usr/local/lib/R/site-library/StanHeaders/include/src/"  -I"/usr/local/lib/R/site-library/StanHeaders/include/"  -I"/usr/local/lib/R/site-library/RcppParallel/include/"  -I"/usr/local/lib/R/site-library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fpic  -g -O2 -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g  -c foo.c -o foo.o
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:88,
##                  from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name ‘namespace’
##   628 | namespace Eigen {
##       | ^~~~~~~~~
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
##   628 | namespace Eigen {
##       |                 ^
## In file included from /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Dense:1,
##                  from /usr/local/lib/R/site-library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13,
##                  from <command-line>:
## /usr/local/lib/R/site-library/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: No such file or directory
##    96 | #include <complex>
##       |          ^~~~~~~~~
## compilation terminated.
## make: *** [/usr/local/lib/R/etc/Makeconf:168: foo.o] Error 1

2. Coeficientes

br_vs_br_coeficients(bayesion_model_1, bayesion_model_5, params_5)

3. Validación

models_validation(lineal_model_1, bayesion_model_1, params_1, vars_1, test_set)

4. Validacion

vars_5 <- c('bill_length_mm', 'bill_depth_mm', 'flipper_length_mm') 

models_validation(lineal_model_1, bayesion_model_5, params_5, vars_5, test_set)
bayesion_predictor_5 <- BayesianRegressionPredictor.from(bayesion_model_5, params_5, vars_5)

plot_compare_fit(
  bayesion_predictor_1,
  bayesion_predictor_5,
  train_set,
  label_1='Regresion Bayesiana con dist informativa', 
  label_2='Regresion Bayesiana con dist menos informativa'
)

2. Modelo bayesiano con parametro con distribucion muy informativa sesgada o con poca varianza

COMPLETAR